Solution — 2024 Question 8 Probability

(a)

\begin{matrix} P (B) & = P (A \cup B) - P (A \cap B^{'}) \\ = 0.51 - 0.24 \\ = 0.27 ∎ \end{matrix}

Use a Venn diagram to see why

(b)

\begin{matrix} P (A \cap B) & = P (A) - P (A \cap B^{'}) \\ = 0.3 - 0.24 \\ = 0.06 \end{matrix}

\begin{matrix} P (A) P (B) & = 0.3 (0.27) \\ = 0.081 \end{matrix}

Since $P (A \cap B) \neq P (A) P (B),$ $A$ and $B$ are not independent $∎$